11,689
pages

The N-growing hierarchy is a hierarchy/notation based on the fast-growing hierarchy, created by Japanese googologist Aeton (2013) [1].

## Definition

• $$[m]_0(n) = m\times n$$
• $$[m]_{\alpha+1}(n) = [m]^{n-1}_\alpha(m)$$, and if $$n=1$$, $$[m]_{\alpha+1}(1)=[m]^0_\alpha(m)=m$$
• $$[m]_\alpha(n) = [m]_{\alpha[n]}(m)$$, when $$\alpha$$ is a limit ordinal and $$\alpha[n]$$ is the $$n$$th term of fundamental sequence assigned to ordinal $$\alpha$$.

And when $$m=10$$, it can be called 10-growing hierarchy. And similarly, 3-growing hierarchy, 16-growing hierarchy, or Googol-growing hierarchy are also possible.

However, If $$m=n$$, it is called Diagonal n-growing hierarchy and its notation changes as follows.

• $$(N_\alpha(n) = [n]_\alpha(n))$$
• $$N_0(n) = n\times n=n^2$$
• $$N_{\alpha+1}(n) = N^{n-1}_\alpha(n)$$
• $$N_\alpha(n) = N_{\alpha[n]}(n)$$

### Examples

This function is exactly equal to up-arrow notation, and probably array notation, but for that reason, when $$m=2$$ and $$\alpha\geq\omega$$, it does not grow well.

• $$[16]_4(8) = 16\uparrow^4 8$$
• $$[10]_{\omega+1}(100) = \{10,100,1,2\}=$$ Corporal
• $$[3]^{64}_{\omega}(4)$$ = Graham's number $$\lesssim[4]_{\omega+1}(65) = \{4,65,1,2\}$$
• $$[4]_{\omega^2+1}(64) = \{4,64,1,1,2\}<$$ Fish number 1
• $$N_\omega(3) = [3]_3(3) = 3\uparrow^3 3=$$ Tritri
• $$N_{\omega^2}(10) = \{10,10,10,10\}=$$ General

Because of the reason that $$[m]_{\omega^\omega}(n)=\{m,n+2(1)2\}$$, this function doesn't match exactly over $$\{m,n(1)2\}$$ level of BEAF, in $$\alpha\geq\omega^\omega$$ level.

• $$N_{\omega^{98}}(10) = [10]_{\omega^\omega}(98) = \{10,100 (1) 2\}=$$ Goobol
• $$[10]_{\omega^\omega}([10]_{\omega^\omega}(98)-2)=$$ goobolplex $$\approx[10]^2_{\omega^\omega}(98)$$

## Sources

1. n-growing hierarchy (Japanese Page)

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea

Methodology

By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system

Proofs, translation maps for analysis schema, and other mathematical contributions

By ふぃっしゅ (Fish): Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's $$\mathcal{O}$$
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the $$\Sigma_1$$-soundness and the pointwise well-definedness of $$\textrm{TR}(T,n)$$ for the case where $$T$$ is the formalisation of the base theory

Entertainments

By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud